Finite Group Databasebeta

← #27287230

Subgroups — #27287230

Order 13568.

402 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20278020C211yesmaximal, characteristic, complemented
2#20273210C212yesmaximal
3#20272616C212yesmaximal, complemented
4#20272544C211yesmaximal, characteristic, complemented
5C53×#1963C211yesmaximal, characteristic
6#16810676C2212yes—
7#16810565—22no—
8#16810557—22no—
9C4×#11719919—22no—
10C4×C4⋊Dic53C2212yes—
11#16810357—22no—
12#16810357—22no—
13C53×#532C2211yescharacteristic
14C53×#423C2211yescharacteristic
15C2×#104×C53—22no—
16C53×#376C2211yescharacteristic
17C2×C4⋊Dic53—22no—
18C2×#11719919—22no—
19C2×#11719919—22no—
20C2×C4×Dic53—22no—
21C4⋊#88322—22no—
22C22⋊#88322—22no—
23C4×#88322—22no—
24#13133124—22no—
25#13133124—22no—
26#13133124D412yes—
27#13133124D412yes—
28C2×#41×C53—21no—
29C2×M16×C53—21no—
30C4⋊Q8×C53Q811yescharacteristic
31#127×C53Q811yescharacteristic
32#124×C53—21no—
33#121×C53—22no—
34C4×D4×C53—22no—
35C2×C42×C53D412yes—
36C2×C42×C53C2311yescharacteristic
37C4⋊C8×C53—22no—
38#104×C53—44no—
39#95×C53—22no—
40#41×C53—42no—
41#41×C53—42no—
42C2×Q8×C53—21no—
43C2×D4×C53—21no—
44C2×D4×C53—22no—
45C2×D4×C53—41no—
46C22×C4×C53—22no—
47C22×C4×C53—41no—
48C22×C4×C53C2×D412yes—
49C22×C4×C53C2×D411yescharacteristic
50C22×Dic53—22no—
51C22×Dic53—22no—
52M16×C53—42no—
53C2×C8×C53—41no—
54C4⋊C4×C53—22no—
55C22⋊C4×C53—42no—
56C42×C53—22no—
57C42×C53—22no—
58C42×C53C2×Q811yescharacteristic
59C42×C53#4111yescharacteristic
60C42×C53#4112yes—
61C4⋊Dic53—44no—
62#11719919—44no—
63#11719919—44no—
64C4×Dic53—42no—
65C4×Dic53—42no—
66C2×#88322—42no—
67C23×C53—41no—
68C23×C53#12011yescharacteristic
69Q8×C53—42no—
70D4×C53—42no—
71D4×C53—82no—
72D4×C53—82no—
73C2×C4×C53—42no—
74C2×C4×C53—42no—
75C2×C4×C53—42no—
76C2×C4×C53—41no—
77C2×C4×C53—42no—
78C2×C4×C53—81no—
79C2×C4×C53—81no—
80C2×C4×C53C22⋊Q812yes—
81C2×C4×C53C22⋊Q811yescharacteristic
82C2×C4×C53#12412yes—
83C2×C4×C53#12411yescharacteristic
84C2×C4×C53#13512yes—
85C2×C4×C53AGL(1,ℤ/8)12yes—
86C2×Dic53—24no—
87C2×Dic53—42no—
88C2×Dic53—42no—
89C2×Dic53—42no—
90C2×Dic53—42no—
91C2×Dic53—42no—
92C8×C53—81no—
93#88322—82no—
94#10372—531nomaximal, Sylow
95C22×C53—22no—
96C22×C53—81no—
97C22×C53—82no—
98C22×C53#39311yescharacteristic
99C22×C53#44912yes—
100C4×C53—22no—
101C4×C53—22no—
102C4×C53—82no—
103C4×C53—81no—
104C4×C53#47812yes—
105C4×C53#48811yescharacteristic
106C4×C53#49011yescharacteristic
107Dic53—44no—
108Dic53—82no—
109#2402—531no—
110#1963D5311yescharacteristic
111#1502—532no—
112#1303—532no—
113#1274—531no—
114C2×C53—22no—
115C2×C53—161no—
116C2×C53#202511yescharacteristic
117C2×C53#203112yes—
118#532C2×D5311yescharacteristic
119#431—532no—
120#423C2×D5311yescharacteristic
121C2×#104—22no—
122#400—1062no—
123#398—1062no—
124C4×C4⋊C4—532no—
125C4×C4⋊C4—1062no—
126#376C2×D5311yescharacteristic
127#327—1062no—
128#327—1062no—
129C53#1037211yescharacteristic, complemented, Sylow
130C2×#41—21no—
131C2×M16—21no—
132C4⋊Q8Dic10611yescharacteristic
133#127Dic10611yescharacteristic
134#124—21no—
135#121—22no—
136C4×D4—22no—
137C2×C4⋊C4—1062no—
138C2×C4⋊C4—1062no—
139C2×C4⋊C4—1062no—
140C2×C42—1062no—
141C2×C42C22⋊D5312yes—
142C2×C42C22×D5311yescharacteristic
143C4⋊C8—22no—
144C4⋊C8—1062no—
145#104—44no—
146C22⋊C8—1062no—
147C4×C8—1062no—
148#95—22no—
149#95—532no—
150#95—532no—
151#95—1062no—
152#95—1062no—
153#41—42no—
154#41—42no—
155C2×Q8—21no—
156C2×D4—22no—
157C2×D4—21no—
158C2×D4—41no—
159C22×C4—22no—
160C22×C4—41no—
161C22×C4—1062no—
162C22×C4—1062no—
163C22×C4D4×D5312yes—
164C22×C4C2×C22⋊D5311yescharacteristic
165M16—42no—
166C2×C8—41no—
167C2×C8—2122no—
168C4⋊C4—22no—
169C4⋊C4—2124no—
170C4⋊C4—2124no—
171C4⋊C4—2124no—
172C22⋊C4—42no—
173C42—22no—
174C42—22no—
175C42—2122no—
176C42—2122no—
177C42C2×Dic10611yescharacteristic
178C42#1171994511yescharacteristic
179C42#1171994712yes—
180C23—41no—
181C23#1313318511yescharacteristic
182Q8—42no—
183D4—42no—
184D4—82no—
185D4—82no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—1064no—
194C2×C4—2122no—
195C2×C4—2122no—
196C2×C4—2122no—
197C2×C4—2122no—
198C2×C4—2122no—
199C2×C4#1313319412yes—
200C2×C4#1313319611yescharacteristic
201C2×C4#1313320312yes—
202C2×C4#1313320411yescharacteristic
203C2×C4#1313323712yes—
204C2×C4#1313324312yes—
205C8—81no—
206C8—4242no—
207C22—22no—
208C22—81no—
209C22—82no—
210C22#1681053511yescharacteristic
211C22#1681076612yes—
212C4—22no—
213C4—22no—
214C4—82no—
215C4—81no—
216C4—4242no—
217C4—2124no—
218C4#1681092312yes—
219C4#1681097711yescharacteristic
220C4#1681098711yescharacteristic
221C2—22no—
222C2—161no—
223C2#2027586111yescharacteristic
224C2#2027588512yes—
subgroupG/H
centerC2×C4#13133204
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1963C2
socleC22×C53#393
prime radicalC53×#532C22
hypercenter#1963D53
hypocenterC53#10372
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1963

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20278020C2semidirect
3. #20272616C2semidirect
4. #20272544C2semidirect
129. C53#10372semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202780202
#202732102
#202726162
#202725442
C53×#19632
#1037253